Robust Mediators in Large Games∗ Michael Kearns† Mallesh M. Pai‡ Ryan Rogers§ Aaron Roth¶ Jonathan Ullmank December 14, 2015 Abstract A mediator is a mechanism that can only suggest actions to players, as a function of all agents’ reported types, in a given game of incomplete information. We study what is achievable by two kinds of mediators, “strong” and “weak.” Players can choose to opt-out of using a strong mediator but cannot misrepresent their type if they opt-in. Such a mediator is “strong” because we can view it as having the ability to verify player types. Weak mediators lack this ability— players are free to misrepresent their type to a weak mediator. We show a striking result—in a prior-free setting, assuming only that the game is large and players have private types, strong mediators can implement approximate equilibria of the complete-information game. If the game is a congestion game, then the same result holds using only weak mediators. Our result follows from a novel application of differential privacy, in particular, a variant we propose called joint differential privacy. arXiv:1512.02698v2 [cs.GT] 11 Dec 2015 ∗We gratefully acknowledge the support of NSF Grant CCF-1101389 and the Alfred P. Sloan Foundation. We thank Nabil Al-Najjar, Eduardo Azevdeo, Eric Budish, Tymofiy Mylovanov, Andy Postlewaite, Al Roth, Tim Roughgarden, Ilya Segal, and Rakesh Vohra for helpful comments and discussions. †Department of Computer and Information Science, University of Pennsylvania. ‡Department of Economics, University of Pennsylvania. §Department of Applied Mathematics and Computational Sciences, University of Pennsylvania. ¶Department of Computer and Information Science, University of Pennsylvania. kCollege of Computer and Information Science, Northeastern University. Most of this work was done while the author was in the School of Engineering and Applied Sciences at Harvard University. 1 Contents 1 Introduction 3 2 Model and Preliminaries 8 3 Joint Differential Privacy and Incentives 14 4 Weak Mediators for Congestion Games 17 5 Strong Mediators for General Games 24 6 Discussion 32 A Privacy Preliminaries 40 B Binary Mechanism 41 C Omitted Proofs from Weak Mediators Section 42 D Proofs of Noise Tolerance of No Regret Algorithms (Section 5.2) 47 E Proofs for Computing Equilibria in Games with Few Actions (Section 5.3.1) 49 F Proof of the Lower Bound on Error (Theorem 13) 51 G Computing Equilibria in Games with Many Actions 55 H Additional Related Work 59 2 1 Introduction Mechanism design generally supposes a principal who can greatly influence the “rules of the game” among agents, to achieve a desirable outcome. In several applied settings of interest, however, this sort of influence is too much to hope for. For example, a market designer may have to live within the constraints of an existing market. She may be unable to guarantee that players participate in her designed mechanism, or that they follow the outcomes proposed even if they do participate. Similarly, in many settings, transfers may be ruled out.1 To distinguish these principals of limited power from the standard principals considered in the literature, we refer to them as “mediators.” In this paper, we study what sorts of outcomes are achievable by mediators. To be more precise, given a game of incomplete information, a mediator may only collect reports from players, and then suggest actions to each player. That is, we augment a given game with an additional option for each player to use a mediator. If a player opts in to using the mediator, and reveals her type, then it will suggest an action for that player to take (as a possibly stochastic function of all reports it received). We further require that mediators be robust or prior-free, i.e. that the mediator’s recommendations (and the incentives of players to follow them) do not rely on the knowledge (or even existence) of any prior distribution over types. Our main results are to show that it is possible to construct a robust mediator under very mild assumptions, namely, that the game is large and that the setting is one of private values. For every profile of realized types, our mediator suggests to each player her action in an approximate equilibrium of the realized full-information game. Further, we show that opting in to using the mediator, and following its recommendation forms an approximate ex-post Nash equilibrium of the mediated game. The error of the approximation vanishes as the number of players grows. The techniques used in our construction may be of independent interest. We propose and use a novel variant of the influential notion of differential privacy, which we term joint differential privacy. In particular, we devise an algorithm for computing equilibria of a full- information game such that any single player’s reported type to the algorithm only has a small effect on the distribution of suggested actions to all other players, in the worst case over the reports of all players (it is this worst case guarantee that yields prior-free robustness). 1A leading example is the central matching clearinghouses proposed for matching markets such as school choice, residency markets etc. Agents are free to drop out and contract on their own before the match, or indeed may deviate from the proposed match, and there are no transfers. A more recent example, closer to our model concerns the design of traffic-aware smartphone routing apps. Again, agents are free to not use the app, or may even try to mislead the app to get favorable routes. For a recent case study, see http://www.wsj.com/articles/in-l-a-one-way-to-beat-traffic-runs-into-backlash-1447469058. 3 Ideas from the differential privacy literature allow us to precisely control the amount by which any single agent’s type influences the suggested actions to others, and hence allows us to ensure our desired incentive properties. In other words, we prove our incentive guarantees for our mechanism by showing that it is differentially private, a technique that may be useful in other settings. In this paper, we consider two kinds of mediators, which we call weak, and strong me- diators, that extend the original game. For both types of mediators, players may simply opt out of using the mediator and play the original game using any strategy they choose. Similarly, for both types of mediators, if they opt in to using the mediator, they are under no obligation to follow the action suggested to them, and can use the suggestion to act however they like. If the mediator is weak, then agents can also arbitrarily misrepresent their type to the mediator. In contrast, if players opt in to using a strong mediator, then they cannot misrepresent their type (equivalently, strong mediators have the ability to verify a player’s type given that she opted in, which is why we term them “strong”). In all cases, we assume that the mediator has very limited power—it cannot modify payoffs of the game (i.e. make payments) or enforce that any player take any particular action. Under only the assumptions that the original game is “large,” and that types are private, we show that it is possible to implement a correlated equilibrium of the complete-information game using a strong mediator.2 Informally, a game is large if there are many players and any player individually has only a small effect on the utility of any other player. Remarkably, no further assumptions on the game (e.g. single crossing properties) are required. We show that in such games there exists a strong mediator such that it is an approximate ex-post Nash equilibrium for every player to opt in to using the mediator, and then faithfully follow the suggested action of the mediator. In particular, this behavior forms a Bayes-Nash equilibrium for any prior on agent types, since it is an ex-post Nash equilibrium. Moreover, when players follow this strategy, the resulting play forms an approximate correlated equilibrium of the complete-information game. We also show that for more structured large games—in particular, congestion games—it is possible to implement an approximate Nash equilibrium of the complete-information game using only a weak mediator. Congestion games are examples of potential games, introduced by Monderer & Shapley (1996). They describe them thus: “The class of congestion games is, on the one hand, narrow, but on the other hand, very important for economics. Any game where a collection of homo- geneous agents have to choose from a finite set of alternatives, and where the 2To be precise, we only guarantee that our mediator will select some approximate equilibrium of the realized continuum, not that the designer has control over what equilibrium is selected. 4 payoff of a player depends on the number of players choosing each alternative, is a congestion game.” (Monderer & Shapley 1996) In fact, they show that the class of potential games is isomorphic to the class of congestion games.3 For congestion games, we construct a weak mediator such that it is an approximate ex-post Nash equilibrium for every player to opt into using the mediator, truthfully report their type, and then faithfully follow the mediator’s suggested action, and when players do so, the resulting play forms an approximate Nash equilibrium of the complete-information game. In both of these results, the approximation error tends to 0 as the number of players grows. A tempting approach to obtaining our results is to consider the following mediator. The mediator accepts a report of each agent’s type, which defines an instance of a complete- information game. It then computes an equilibrium (either Nash or correlated) of the full- information game, and suggests an action to each player which is a draw from this equi- librium distribution.